1. **Problem:** Solve the quadratic equation $3x^2 - 5x - 9 = 0$ using the quadratic formula and give answers correct to 2 decimal places. 2. **Formula:** The quadratic formula is given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. **Identify coefficients:** For $3x^2 - 5x - 9 = 0$, we have $a=3$, $b=-5$, and $c=-9$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 3 \times (-9) = 25 + 108 = 133$$ 5. **Calculate the roots:** $$x = \frac{-(-5) \pm \sqrt{133}}{2 \times 3} = \frac{5 \pm \sqrt{133}}{6}$$ 6. **Evaluate the square root:** $$\sqrt{133} \approx 11.53$$ 7. **Find the two solutions:** - First root: $$x_1 = \frac{5 + 11.53}{6} = \frac{16.53}{6} \approx 2.76$$ - Second root: $$x_2 = \frac{5 - 11.53}{6} = \frac{-6.53}{6} \approx -1.09$$ **Final answer:** $x \approx 2.76$ or $x \approx -1.09$